Title:

The geometrical thought of Isaac Newton : an examination of the meaning of geometry between the 16th and 18th centuries

Our thesis explores aspects of the geometrical work and thought of Isaac Newton in order to better understand and reevaluate his approach to geometry, and specifically his synthetic methods and the organic description of plane curves. In pursuing this research we study Newton's geometrical work in the context of the changing view of geometry between the late 16th and early 18th centuries, a period defined by the responses of the early modern geometers to a new Latin edition of Pappus' Collectio. By identifying some of the major challenges facing geometers of this period as they attempted to define and practice geometry we are able to contrast Newton's own approach to geometry. The themes emerging from the geometrical thought of early modern geometers provide the mathematical context from which to understand, interpret and reevaluate the approach taken by Newton. In particular we focus on Newton's profound rejection of the new algebraic Cartesian methods and geometrical philosophies, and the opportunity to focus more clearly on some of his most astonishing geometrical contributions. Our research highlights Newton's geometrical work and examines specific examples of his synthetic methods. In particular we draw attention to the significance of Newton's organic construction and the limitations of Whiteside's observations on this subject. We propose that Newton's organic rulers were genuinely original. We disagree with Whiteside that they were inspired by van Schooten, except in the loosest sense. Further, we argue that Newton's study of singular points by their resolution was new, and that it has been misunderstood by Whiteside in his interpretation of the transformation effected by the rulers. We instead emphasise that it was the standard quadratic transformation. Overall we wish to make better known the importance of geometry in Newton's scientific thought, as well as highlighting the mathematical and historical importance of his organic description of curves as an example of his synthetic approach to geometry. This adds to contemporary discourse surrounding Newton's geometry, and specifically provides a foundation for further research into the implications of Newton's geometrical methods for his successors.
